We can simplify the expression:

\[
\log \sqrt{65} + \log \sqrt{2} - \log \sqrt{13}
\]

### Step 1: Use the property \(\log \sqrt{a} = \frac{1}{2} \log a\):
\[
\log \sqrt{65} = \frac{1}{2} \log 65
\]
\[
\log \sqrt{2} = \frac{1}{2} \log 2
\]
\[
\log \sqrt{13} = \frac{1}{2} \log 13
\]

### Step 2: Substitute these into the original expression:
\[
\frac{1}{2} \log 65 + \frac{1}{2} \log 2 - \frac{1}{2} \log 13
\]

### Step 3: Factor out \(\frac{1}{2}\):
\[
\frac{1}{2} \left( \log 65 + \log 2 - \log 13 \right)
\]

### Step 4: Use the properties of logarithms:
\[
\log a + \log b = \log (a \times b), \quad \log a - \log b = \log \left( \frac{a}{b} \right)
\]

Thus, we simplify the logs inside the parentheses:
\[
\log 65 + \log 2 = \log (65 \times 2) = \log 130
\]
\[
\log 130 - \log 13 = \log \left( \frac{130}{13} \right) = \log 10
\]

### Step 5: Simplify the expression:
\[
\frac{1}{2} \log 10
\]

Since \(\log 10 = 1\), we get:
\[
\frac{1}{2} \times 1 = \frac{1}{2}
\]

Thus, the simplified value of the expression is:
\[
\boxed{\frac{1}{2}}
\]

Would you like more details on any step?

Here are some related questions for practice:
1. Simplify \(\log \sqrt{50} + \log \sqrt{5} - \log \sqrt{25}\).
2. Evaluate \(\log 200 - \log 2\).
3. Simplify \(\log \sqrt{48} + \log 2 - \log 12\).
4. Use logarithms to simplify \(\log 16 + \log 25 - \log 100\).
5. Simplify \(\log \sqrt{81} + \log \sqrt{4} - \log 9\).

**Tip:** Remember, logarithms turn multiplication into addition and division into subtraction, which can make simplifying expressions easier!

Discover how to simplify the logarithmic expression Log √65 + log √2 - log √13 using properties of logarithms. Follow the step-by-step guide that explains key logarithmic formulas, including the square root and multiplication rules. Suitable for high school students (Grades 9-12).

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